Editing Mean free path (section)

==Scattering theory==
<!-- also, inconsistent notation re probability density or CDF. -->
[[File:Mean free path.png|frame|Slab of target]]
Imagine a beam of particles being shot through a target, and consider an infinitesimally thin slab of the target (see the figure).<ref>{{cite book |last1=Chen |first1=Frank F.|title=Introduction to Plasma Physics and Controlled Fusion |date=1984 |publisher=Plenum Press |isbn=0-306-41332-9 |page=156 |edition=1st}}</ref>  The atoms (or particles) that might stop a beam particle are shown in red. The magnitude of the mean free path depends on the characteristics of the system. Assuming that all the target particles are at rest but only the beam particle is moving, that gives an expression for the mean free path:

:<math>\ell = (\sigma n)^{-1},</math>

where {{mvar|ℓ}} is the mean free path, {{mvar|n}} is the number of target particles per unit volume, and {{mvar|&sigma;}} is the effective [[cross section (physics)|cross-sectional]] area for collision.

The area of the slab is {{math|''L''<sup>2</sup>}}, and its volume is {{math|''L''<sup>2</sup> ''dx''}}.  The typical number of stopping atoms in the slab is the concentration {{mvar|n}} times the volume, i.e., {{math|''n L''<sup>2</sup> ''dx''}}.  The probability that a beam particle will be stopped in that slab is the net area of the stopping atoms divided by the total area of the slab:

:<math>\mathcal{P}(\text{stopping within }dx) = \frac{\text{Area}_\text{atoms}}{\text{Area}_\text{slab}} = \frac{\sigma n L^{2}\, dx}{L^{2}} = n \sigma\, dx,</math>

where {{mvar|&sigma;}} is the area (or, more formally, the "[[scattering cross-section]]") of one atom.

The drop in beam intensity equals the incoming beam intensity multiplied by the probability of the particle being stopped within the slab:

:<math>dI = -I n \sigma \,dx.</math>

This is an [[ordinary differential equation]]:

:<math>\frac{dI}{dx} = -I n \sigma \overset{\text{def}}{=} -\frac{I}{\ell},</math>

whose solution is known as [[Beer–Lambert law]] and has the form <math>I = I_{0} e^{-x/\ell}</math>, where {{mvar|x}} is the distance traveled by the beam through the target, and {{math|''I''<sub>0</sub>}} is the beam intensity before it entered the target; {{mvar|ℓ}} is called the mean free path because it equals the [[mean]] distance traveled by a beam particle before being stopped. To see this, note that the probability that a particle is absorbed between {{mvar|x}} and {{math|''x'' + ''dx''}} is given by

:<math>d\mathcal{P}(x) = \frac{I(x)-I(x+dx)}{I_0} = \frac{1}{\ell} e^{-x/\ell} dx.</math>

Thus the [[expectation value]] (or average, or simply mean) of {{mvar|x}} is

:<math>\langle x \rangle \overset{\text{def}}{=} \int_0^\infty x d\mathcal{P}(x) = \int_0^\infty \frac{x}{\ell} e^{-x/\ell} \, dx = \ell.</math>

The fraction of particles that are not stopped ([[attenuation|attenuated]]) by the slab is called [[Transmittance|transmission]] <math>T = I/I_{0} = e^{-x/\ell}</math>, where {{mvar|x}} is equal to the thickness of the slab.